Multi-Dimensional Sigma-Functions

Abstract

In 1997 the present authors published a review (Ref. BEL97 in the present manuscript) that recapitulated and developed classical theory of Abelian functions realized in terms of multi-dimensional sigma-functions. This approach originated by K.Weierstrass and F.Klein was aimed to extend to higher genera Weierstrass theory of elliptic functions based on the Weierstrass $\sigma$-functions. Our development was motivated by the recent achievements of mathematical physics and theory of integrable systems that were based of the results of classical theory of multi-dimensional theta functions. Both theta and sigma-functions are integer and quasi-periodic functions, but worth to remark the fundamental difference between them. While theta-function are defined in the terms of the Riemann period matrix, the sigma-function can be constructed by coefficients of polynomial defining the curve. Note that the relation between periods and coefficients of polynomials defining the curve is transcendental. Since the publication of our 1997-review a lot of new results in this area appeared (see below the list of Recent References), that promoted us to submit this draft to ArXiv without waiting publication a well-prepared book. We complemented the review by the list of articles that were published after 1997 year to develop the theory of $\sigma$-functions presented here. Although the main body of this review is devoted to hyperelliptic functions the method can be extended to an arbitrary algebraic curve and new material that we added in the cases when the opposite is not stated does not suppose hyperellipticity of the curve considered.

267 pages, 4 figures

References in corpus (6)

• Analytical solution of the geodesic equation in Kerr-(anti) de Sitter space-times
• Analytic treatment of complete and incomplete geodesics in Taub-NUT space-times
• Hyperelliptic Solutions of KdV and KP equations: Reevaluation of Baker's Study on Hyperelliptic Sigma Functions
• Hyperelliptic Loop Solitons with Genus g: Investigations of a Quantized Elastica
• Integrable Systems with Pairwise Interactions and Functional Equations
• Closed Loop Solitons and Sigma Functions: Classical and Quantized Elasticas with Genera One and Two

Cited by in corpus (15)

• Differentiation of Genus 3 Hyperelliptic Functions
• On Addition Formulae for Sigma Functions of Telescopic Curves
• Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves
• Parametric Korteweg--de Vries hierarchy and hyperelliptic sigma functions
• Construction of Two Parametric Deformation of KdV-Hierarchy and Solution in Terms of Meromorphic Functions on the Sigma Divisor of a Hyperelliptic Curve of Genus 3
• On Jacobi Inversion Formulae for Telescopic Curves
• Relationships Between Hyperelliptic Functions of Genus 2 and Elliptic Functions
• Hyperelliptic sigma functions and the Kadomtsev-Petviashvili equation
• Notes about a combinatorial expression of the fundamental second kind differential on an algebraic curve
• Analytical and number-theoretical properties of the two-dimensional sigma function
• Eñe product in the transalgebraic class
• The Half-period Addition Formulae for Genus Two Hyperelliptic $\wp$ Functions and the Sp(4,$\mathbb{R}$) Lie Group Structure
• On Regularization of Second Kind Integrals
• The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications
• Periods of second kind differentials of (n,s)-curves